In computer science and mathematics, expressions can be represented and manipulated in different formats, such as infix, prefix (also known as Polish notation), and postfix (also known as Reverse Polish Notation or RPN). Each format has its advantages and use cases, and understanding all three can be beneficial for various reasons:
Prefix Notation (Polish Notation):
In prefix notation, operators precede their operands. For example, the infix expression A + B would be written as + A B in prefix notation. Easier to parse and evaluate algorithmically: Prefix notation eliminates the need for parentheses and rules for operator precedence (like PEMDAS/BODMAS). This simplifies parsing and evaluation algorithms, making it efficient for computer processing. Suitable for stack-based algorithms: Prefix notation is well-suited for stack-based algorithms such as expression evaluation using stacks or recursive algorithms.
Postfix Notation (Reverse Polish Notation – RPN):
In postfix notation, operators follow their operands. For example, the infix expression A + B would be written as A B + in postfix notation.Eliminates ambiguity: Postfix notation removes the need for parentheses and ambiguity regarding operator precedence because each operator immediately follows its operands. Efficient for stack-based processing: Postfix notation is particularly efficient for stack-based processing, making it suitable for calculator applications and mathematical evaluations.
In infix notation, operators are placed between their operands. This is the conventional mathematical notation most humans are familiar with.
Familiarity: Infix notation is the standard notation used in mathematics and is more intuitive for many people when reading and writing mathematical expressions.
Ease of comprehension: In many cases, infix notation can be easier to read and understand for humans, especially in simple arithmetic expressions.
Why We Need Prefix and Postfix Notations?
Efficient Parsing and Evaluation: Prefix and postfix notations eliminate the need for complex parsing and precedence rules, making them efficient for computer algorithms and implementations. This efficiency is crucial in programming languages, calculators, compilers, and other systems dealing with mathematical expressions.
Stack-Based Algorithms: Both prefix and postfix notations are well-suited for stack-based algorithms, which are commonly used in computer science for evaluating expressions, parsing, and implementing calculators.
Eliminating Ambiguity: Prefix and postfix notations remove ambiguity related to operator precedence and the need for parentheses, simplifying expression evaluation and processing.
Suitability for Automated Processing: Prefix and postfix notations are easier to process algorithmically, making them suitable for automated processing in computer programs and systems.
Overall, while infix notation is familiar and intuitive for humans, prefix and postfix notations offer efficiency, simplicity in parsing, and suitability for stack-based processing, making them valuable in various computer science applications.
how to convert the infix expression to postfix and prefix notations?
To convert an infix expression to postfix or prefix notation, you can use the algorithmic approach known as the Shunting Yard algorithm. This algorithm was developed by Edsger Dijkstra and is widely used for parsing mathematical expressions. Here’s a step-by-step guide to converting an infix expression to postfix and prefix notations using the Shunting Yard algorithm:
Infix to Postfix Conversion:
Initialize an empty stack and an empty string to store the postfix expression.
Scan the infix expression from left to right.
For each token in the expression:
- If the token is an operand (number or variable), add it to the postfix string.
- If the token is an operator:
- While the stack is not empty and the top of the stack has higher precedence than or equal precedence to the current token, pop operators from the stack and add them to the postfix string.
- Push the current token onto the stack.
- If the token is a left parenthesis ‘(‘, push it onto the stack.
- If the token is a right parenthesis ‘)’:
- Pop operators from the stack and add them to the postfix string until a left parenthesis is encountered (do not include the left parenthesis in the output).
- Discard the left parenthesis from the stack.
After scanning all tokens, pop any remaining operators from the stack and add them to the postfix string.
The resulting string is the postfix notation of the original infix expression.
function infixToPostfix(expression) {
const precedence = {
'+': 1,
'-': 1,
'*': 2,
'/': 2,
'^': 3
};
let postfix = '';
const stack = [];
for (let token of expression.split(' ')) {
if (token.match(/[A-Za-z0-9]/)) {
postfix += token + ' ';
} else if (token === '(') {
stack.push(token);
} else if (token === ')') {
while (stack.length && stack[stack.length - 1] !== '(') {
postfix += stack.pop() + ' ';
}
stack.pop(); // Discard the '('
} else { // Operator
while (stack.length && precedence[token] <= precedence[stack[stack.length - 1]]) {
postfix += stack.pop() + ' ';
}
stack.push(token);
}
}
while (stack.length) {
postfix += stack.pop() + ' ';
}
return postfix.trim();
}
// Example usage:
const infixExpression = 'A + B * (C - D) / E';
const postfixExpression = infixToPostfix(infixExpression);
console.log('Postfix Expression:', postfixExpression); // Output: ABCD-*E/+
Infix to Prefix Conversion:
To convert an infix expression to prefix notation, you can follow a similar approach as for postfix conversion, but with a few modifications:
Reverse the infix expression.
Initialize an empty stack and an empty string to store the prefix expression.
Scan the reversed infix expression from left to right.
For each token in the reversed expression:
- If the token is an operand, add it to the prefix string.
- If the token is an operator:
- While the stack is not empty and the top of the stack has higher precedence than or equal precedence to the current token, pop operators from the stack and add them to the prefix string.
- Push the current token onto the stack.
- If the token is a right parenthesis ‘)’, push it onto the stack.
- If the token is a left parenthesis ‘(‘:
- Pop operators from the stack and add them to the prefix string until a right parenthesis is encountered (do not include the right parenthesis in the output).
- Discard the right parenthesis from the stack.
After scanning all tokens, pop any remaining operators from the stack and add them to the prefix string.
Reverse the resulting string to get the prefix notation of the original infix expression.
Here’s an example to illustrate the conversion process:
Infix Expression: A + B * (C – D) / E
Postfix Expression: ABCD-E/+ Prefix Expression: +A/B-CDE
You can use these steps and examples to convert other infix expressions to postfix and prefix notations as needed.
function infixToPrefix(expression) {
const precedence = {
'+': 1,
'-': 1,
'*': 2,
'/': 2,
'^': 3
};
const reversedExpression = expression.split('').reverse().join('');
let prefix = '';
const stack = [];
let token = '';
for (let i = 0; i < reversedExpression.length; i++) {
const char = reversedExpression[i];
if (char === ' ') {
continue;
}
if (char.match(/[A-Za-z0-9]/)) {
token = char + token;
if (i === reversedExpression.length - 1 || !reversedExpression[i + 1].match(/[A-Za-z0-9]/)) {
prefix = token + ' ' + prefix;
token = '';
}
} else if (char === ')') {
stack.push(char);
} else if (char === '(') {
while (stack.length && stack[stack.length - 1] !== ')') {
prefix = stack.pop() + ' ' + prefix;
}
stack.pop(); // Discard the ')'
} else { // Operator
while (stack.length && precedence[char] < precedence[stack[stack.length - 1]]) {
prefix = stack.pop() + ' ' + prefix;
}
stack.push(char);
}
}
while (stack.length) {
prefix = stack.pop() + ' ' + prefix;
}
return prefix.trim();
}
// Example usage:
const infixExpression = '(A + B) * (C - D) / E';
const prefixExpression = infixToPrefix(infixExpression);
console.log('Prefix Expression:', prefixExpression); // Output: /*+ABCDE
Prefix expression to function call
As the many computer languages don’t support the complex number directly, the operations of the complex numbers have to be changed into function calls first. Here is the code to change the prefix expression to function call.
function prefixToFunctionCalls(prefixExpression) {
const tokens = prefixExpression.split(' ');
const stack = [];
for (let i = tokens.length - 1; i >= 0; i--) {
const token = tokens[i];
if (token.match(/[A-Za-z0-9]/)) {
stack.push(token);
} else {
const operand1 = stack.pop();
const operand2 = stack.pop();
const functionName = getFunctionName(token);
stack.push(`${functionName}(${operand1},${operand2})`);
}
}
return stack.pop();
}
function getFunctionName(operator) {
switch (operator) {
case '+':
return 'add';
case '-':
return 'sub';
case '*':
return 'mul';
case '/':
return 'div';
default:
throw new Error('Invalid operator');
}
}
// Example usage:
const prefixExpression = "/ * + A B - C D E";
const functionCalls = prefixToFunctionCalls(prefixExpression);
console.log('Function Calls:', functionCalls); // Output: div(mul(add(A,B),sub(C,D)),E)